Designing tasks so that all learners can engage with hard maths Anne Watson Toulouse, 2010

Designing tasks so that all learners can engage with hard

maths

Anne Watson

Toulouse, 2010

Decimals!

• 10% of 232.3

• 20% of 234.6 or 0.23 !!

Teaching context

• All learners generalise all the time• It is the teacher’s role to organise

experience• It is the learners’ role to make sense of

experience

Sorting

2x + 1 3x – 3 2x – 5

x + 1 -x – 5 x – 3

3x + 3 3x – 1 -2x + 1

-x + 2 x + 2 x - 2

Sorting processes

• Sort into two groups – not necessarily equal in size

• Describe the two groups• Now sort the biggest pile into two groups• Describe these two groups• Make a new example for the smallest

groups• Choose one to get rid of which would

make the sorting task different

Sorting grids

+ve y-intercept

-ve y-intercept

Goes through origin

+ve gradient

-ve gradient

Zero gradient

Make your own

• In topics you are currently teaching, what examples could usefully be sorted according to two categories?

Comparing

• In what ways are these pairs the same, and in what ways are they different?

• 4x + 8 and 4(x + 2)• Rectangles and parallelograms

• Which is bigger?• 5/6 or 7/9• A 4 centimetre square or 4 square centimetres

Make your own

• Find two very ‘similar’ things in a topic you are currently teaching which can be usefully compared

• Find two very different things which can be usefully compared

Ordering

• Put these in increasing order:

6√2 4√3 2√8 2√9 9 4√4

Make your own

• What calculations do your students need to practise? Can you construct examples so that the size of the answers is interesting?

Arguing about

• Anne says that when a percentage goes down, the actual number goes down

- Is this always, sometimes or never true?

• John says that when you square a number, the result is always bigger than the number you started with

- Is this always, sometimes or never true?

Make your own

• What assumptions do your students make? What statements could they argue about?

Characterising

• Which multiples of 3 are also square numbers?

• Which quadratic curves go through (0,0)?

• What cubics have coincident roots?

• What angles have interesting trig ratios?

Make your own

• By asking non-standard questions about standard topics, can you get students to practise, and fiddle around with ideas, but with a further purpose?

Construct a ... polygon with

1 2 3 4 5 6

1

2

3

4

5

6

pairs of parallel sides

right

ang

les

Constructing

• Unexpected objects• Unusual objects• Impossible objects

– Brings students face-to-face with the limitations and possibilities of concepts

Make your own

Enlargement (1)

Enlargement (2)

Enlargement (3)

Enlargement (4)

Make your own

• What techniques are you currently teaching? Can you lead your students to understand when they need to give up intuitive methods and adopt more powerful techniques?